3.378 \(\int \frac{e^{-2 i \tan ^{-1}(a x)} x^2}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]

[Out]

(I - 2*a*x)/(6*a^3*c^3*(1 - I*a*x)*(1 + I*a*x)^3)

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Rubi [A]  time = 0.0774534, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5082, 81} \[ \frac{-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(E^((2*I)*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

(I - 2*a*x)/(6*a^3*c^3*(1 - I*a*x)*(1 + I*a*x)^3)

Rule 5082

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rule 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{-2 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^2}{(1-i a x)^2 (1+i a x)^4} \, dx}{c^3}\\ &=\frac{i-2 a x}{6 a^3 c^3 (1-i a x) (1+i a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.032676, size = 36, normalized size = 0.95 \[ \frac{2 a x-i}{6 a^3 c^3 (a x-i)^3 (a x+i)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(E^((2*I)*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]

[Out]

(-I + 2*a*x)/(6*a^3*c^3*(-I + a*x)^3*(I + a*x))

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Maple [A]  time = 0.053, size = 62, normalized size = 1.6 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{-{\frac{i}{8}}}{{a}^{3} \left ( -ax+i \right ) ^{2}}}-{\frac{1}{12\,{a}^{3} \left ( -ax+i \right ) ^{3}}}-{\frac{1}{16\,{a}^{3} \left ( -ax+i \right ) }}-{\frac{1}{16\,{a}^{3} \left ( ax+i \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+I*a*x)^2*(a^2*x^2+1)/(a^2*c*x^2+c)^3,x)

[Out]

1/c^3*(-1/8*I/a^3/(-a*x+I)^2-1/12/a^3/(-a*x+I)^3-1/16/a^3/(-a*x+I)-1/16/a^3/(a*x+I))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^2*(a^2*x^2+1)/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.13267, size = 104, normalized size = 2.74 \begin{align*} \frac{2 \, a x - i}{6 \, a^{7} c^{3} x^{4} - 12 i \, a^{6} c^{3} x^{3} - 12 i \, a^{4} c^{3} x - 6 \, a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^2*(a^2*x^2+1)/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

(2*a*x - I)/(6*a^7*c^3*x^4 - 12*I*a^6*c^3*x^3 - 12*I*a^4*c^3*x - 6*a^3*c^3)

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Sympy [A]  time = 0.8945, size = 56, normalized size = 1.47 \begin{align*} \frac{2 a^{9} x - i a^{8}}{6 a^{15} c^{3} x^{4} - 12 i a^{14} c^{3} x^{3} - 12 i a^{12} c^{3} x - 6 a^{11} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(1+I*a*x)**2*(a**2*x**2+1)/(a**2*c*x**2+c)**3,x)

[Out]

(2*a**9*x - I*a**8)/(6*a**15*c**3*x**4 - 12*I*a**14*c**3*x**3 - 12*I*a**12*c**3*x - 6*a**11*c**3)

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Giac [B]  time = 1.1234, size = 119, normalized size = 3.13 \begin{align*} \frac{1}{32 \, a^{3} c^{3}{\left (i - \frac{2 \, i}{a i x + 1}\right )}} - \frac{\frac{6 \, a^{3} c^{6} i^{3}}{{\left (a i x + 1\right )}^{2}} - \frac{3 \, a^{3} c^{6} i}{a i x + 1} - \frac{4 \, a^{3} c^{6} i^{3}}{{\left (a i x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+I*a*x)^2*(a^2*x^2+1)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

1/32/(a^3*c^3*(i - 2*i/(a*i*x + 1))) - 1/48*(6*a^3*c^6*i^3/(a*i*x + 1)^2 - 3*a^3*c^6*i/(a*i*x + 1) - 4*a^3*c^6
*i^3/(a*i*x + 1)^3)/(a^6*c^9)